Wednesday, August 4, 2010
Russell's Paradox as a Paradox About Properties (The Russell's Paradox Series, Part IV of IV)
Some dialetheists take Russell's Paradox to form the basis of a sound argument for the existence of true contradictions. In Part I of this series, I argued that, given the obvious limitations of our epistemic access to the realm of mathematical objects (if it exists), we should be extremely cautious and conservative in coming to conclusions about its logical structure. In Part II, I argued that, given that our best reason for believing in the existence of any sets comes from Quinean indispensibility considerations, and that we don't need anything even close to the over-populated universe of naive set theory to reconstruct all the mathematics which can't be eliminated from our best science, belief in unrestricted comprehension axiom of naive set theory (and, hence, belief in the existence of the Russell Set) is unmotivated. In Part III, I responded to the best obvious objection--that I was solving Russell's Paradox by appealing to considerations about the epistemology of mathematics that would be obviously inapplicable to the Liar Paradox, and thus solving paradoxes that are 'of a type' in an unacceptable disunified way. I argued that, in fact, given plausible and widely held views about mathematical objects, and Priest's own view about puzzles like "the Barber Paradox," solving Russell's Paradox in a way highly distinct from how one solves the Liar Paradox is actually entirely reasonable.
All of this, however, leaves a large remaining question about Russell's Paradox unanswered. All of my arguments about the rationality of rejecting belief in the Russell Set only solve the paradox if we assume that Russell's Paradox is fundamentally a paradox about sets, i.e. abstract mathematical objects that have other objects as members. Recently, however, Hartry Field and JC Beall, in Saving Truth From Paradox and Spandrels of Truth respectively, have argued that there are two versions of Russell's Paradox--one about sets and the other about properties--and that, even if the former is solvable in straightforwardly orthodox ways, the latter requires some sort of non-classical solution.
In Part I, I said the following:
"JC Beall, if I understand him correctly, follows Hartry Field in separating the paradox into two versions, the traditional set-theoretic one, and a paradox about the property 'does not apply to itself'--Beall accepts classical orthodoxy about the former and handles the latter dialethetically, just as Field accepts classical orthodoxy about the former and handles the latter by denying the relevant instances of the Excluded Middle. To me, the Field/Beall position on the taxonomy of the paradoxes seems bizarre. Russell's Paradox is about sets, and the 'does not apply to itself' paradox is about properties. They have similar structures, but what of it? Lots of paradoxes have Russell's-Paradox-ish structures, and saying that this means that the 'does not apply to itself' paradox is a version of Russell's Paradox strikes me as making no more sense than describing the Liar as 'the sentence version of' Russell's Paradox. (In fact, I'm inclined to think that, in so far as 'applies to' can be paraphrased as something like 'is said of itself in a true sentence', the 'does not apply to itself' paradox has a lot more in common with the Liar than it does with Russell's Paradox.) In both cases, one can talk that way if one wishes, but it doesn't strike me as shedding much light on anything. I think that things are kept clearer by regarding the 'does not apply to itself' paradox as an interesting puzzle in its own right that doesn't have much of anything to do with Russell's Paradox."
Whether or not this view about how to classify the property paradox is correct, since I've been blogging about Russell's Paradox, and some smart big-name philosophers regard the puzzle about properties as a version of Russell's Paradox, I should say something about it here.
My preferred solution would be to say that the predicate "applies to itself" (and, thus, its negation) is only meaningful parasitically. That is to say, for example, the true statement, "the property of 'being a property' applies to itself" means nothing above and beyond "the property of 'being a property' is a property." (Similarly, the false statement, "the property of 'being a property that no one has ever mentioned' applies to itself" means nothing above and beyond "the property of 'being a property that no one has ever mentioned' is a property that no one has ever mentioned.) The phrase 'applies to itself' (or 'does not apply to itself') does not name a distinct property. It is, rather, a convenient linguistic device for referring to other properties. What, after all, could be the extra content of "the property of 'being a property that no one has ever mentioned' applies to itself," above and beyond what's captured by the paraphrase "the property of 'being a property that no one has ever mentioned' is a property that no one has ever mentioned?"
If, however, the meaning of the predicate "applies to itself" is entirely inherited from the meaning of whatever phrase it is appended to, and it has no 'independent' meaning whatsoever, a straightforward consequence of this fact is that the phrase "the property 'does not apply to itself' applies to itself" seems to mean something but is actually meaningless. Ungrounded uses of the predicate "applies to itself" (or, of course, its negation) are as devoid of semantic content as "colorless green ideas sleep furiously."
"Wait a damn second," you might be thinking, "if it's meaningless, how can so many competent speakers of the language think they grasp its meaning? You must mean 'meaningless' in some special technical way that's not how the term is ordinarily used. Also, are you going to try to pull something like this as your solution to the Liar Paradox? If so, aren't you going to have a hell of a hard time getting around revenge paradoxes? What about 'this sentence is either false or meaningless'?"
....to all of which all I can say is "stay tuned for next Monday's post on the Liar Paradox!"